# Polynomials in Standard Form

## Key Questions

• Standard form simply refers to the format of a mathematical expression where the terms are arranged by decreasing order of degree. Where the degree is determined by the exponent value of the variable of each term.

For quadratic equations the standard form is
$a {x}^{2} + b x + c$
Where
$a {x}^{2}$ has a degree of 2
$b x$ has a degree of 1
and $c$ has a degree of zero.

If we took an example like, $- 16 + 5 {f}^{8} - 7 {f}^{3}$
The highest degree is 8 in the term $5 {f}^{8}$
The next degree is 3 in the term $- 7 {f}^{3}$
and the $- 16$ term has a degree of zero because there is no variable.

Therefore the standard form for this expression would be
$5 {f}^{8} - 7 {f}^{3} - 16$

Refer to Explanation.

#### Explanation:

The degree of a polynomial is determined by the highest power of $x$ in the polynomial

e.g.
$f \left(x\right) = 2 {x}^{5} - 3 {x}^{2} + 4$
The highest power of $x$ in f(x) is 5.
Hence, degree of f(x) = 5

A polynomial with only a constant, e.g. $P \left(x\right) = 5$ has a degree of 0 as $5 = 5 {x}^{0}$

• You can simplify polynomials only if they have roots. You can think of polynomials as numbers, and of monomials of the form $\left(x - a\right)$ as prime numbers. So, as you can write a composite numbers as product of primes, you can write a "composite" polynomial as product of monomials of the form $\left(x - a\right)$, where $a$ is a root of the polynomial. If the polynomial has no roots, it means that, in a certain sense, it is "prime", and cannot thus be further simplified.

For example, ${x}^{2} + 1$ has no (real) roots, so it cannot be simplified. On the other hand, ${x}^{2} - 1$ has roots $\setminus \pm 1$, so it can be simplified into $x \left(+ 1\right) \left(x - 1\right)$.
Finally, ${x}^{3} + x$ has a root for $x = 0$. So, we can write as $x \left({x}^{2} + 1\right)$, and for what we saw before, this expression is no longer simplifiable.

• Polynomial Function of Degree n

A polynomial function $f \left(x\right)$ of degree $n$ is of the form

$f \left(x\right) = {a}_{n} {x}^{n} + {a}_{n - 1} {x}^{n - 1} + \cdots + {a}_{1} x + {a}_{0}$,

where ${a}_{n}$ is a nonzero constant, and ${a}_{n - 1} , {a}_{n - 2} , \ldots , {a}_{0}$ are any constants.

Examples

$f \left(x\right) = {x}^{2} + 3 x - 1$ is a polynomial of degree 2, which is also called a quadratic function.

$g \left(x\right) = 2 + x - {x}^{3}$ is a polynomial of degree 3, which is also called a cubic function.

$h \left(x\right) = {x}^{7} - 5 {x}^{4} + {x}^{2} + 4$ is a polynomial of degree 7.

I hope that this was helpful.